Sudden emergence of a giant k-core in a random graph

Boris Pittel, Joel Spencer, Nicholas Wormald

Research output: Contribution to journalArticle

Abstract

The k-core of a graph is the largest subgraph with minimum degree at least k. For the Erdos-Rényi random graph G(n, m) on n vertives, with m edges, it is known that a giant 2-core grows simultaneously with a giant component, that is, when m is close to n/2. We show that for k ≥ 3, with high probability, a giant k-core appears suddenly when m reaches ckn/2; here ck = minλ>0 λ/πk(λ) and πk(λ) = P{Poisson(λ)≥k-1}. In particular, c3≈3.35. We also demonstrate that, unlike the 2-core, when a k-core appears for the first time it is very likely to be giant, of size ≈pkk)n. Here λk is the minimum point of λ/πk(λ) and pkk) = P{Poisson(λk)≥k}. For k = 3, for instance, the newborn 3-core contains about 0.27n vertices. Our proofs are based on the probabilistic analysis of an edge deletion algorithm that always find a k-core if the graph has one.

Original languageEnglish (US)
Pages (from-to)111-151
Number of pages41
JournalJournal of Combinatorial Theory, Series B
Volume67
Issue number1
DOIs
StatePublished - May 1996

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Random Graphs
Siméon Denis Poisson
Giant Component
Probabilistic Analysis
Minimum Degree
Graph in graph theory
Erdös
Deletion
Subgraph
Likely
Demonstrate

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Theoretical Computer Science

Cite this

Sudden emergence of a giant k-core in a random graph. / Pittel, Boris; Spencer, Joel; Wormald, Nicholas.

In: Journal of Combinatorial Theory, Series B, Vol. 67, No. 1, 05.1996, p. 111-151.

Research output: Contribution to journalArticle

Pittel, Boris ; Spencer, Joel ; Wormald, Nicholas. / Sudden emergence of a giant k-core in a random graph. In: Journal of Combinatorial Theory, Series B. 1996 ; Vol. 67, No. 1. pp. 111-151.
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